Conjugacy Distinguished Subgroups

Let $\mathcal{C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually $\mathcal{C}$ group $R$ is said to be conjugacy ${\cal C}$-distinguishedifwhenever$y\in R$,then$y$hasaconjugatein$H$ if and only if the same holds for the images of $y$ and $H$ in every quotient group $R\mathrm{/}N\in \mathcal{C}$ of $R$. We prove that in a group having a normal free subgroup $\mathrm{\Phi }$ such that $R\mathrm{/}\mathrm{\Phi }$ is in $\mathcal{C}$, every finitely generated subgroup is conjugacy $\mathcal{C}$-distinguished. We also prove that finitely generated subgroups of limit groups, of Lyndon groups and certain one-relator groups are conjugacy distinguished ($\mathcal{C}$ here is the class of all finite groups).